Mathematics:

Cohomology
→Theory
→Implementation

Jena:

Faculty

David Green

External links:

Singular

Gap

Mod-3-Cohomology of group number 9 of order 36

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• The group order factors as 2² · 3².
• It is non-abelian.
• It has 3-Rank 2.
• The centre of a Sylow 3-subgroup has rank 2.
• Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  1  −  2·t  +  4·t2  −  4·t3  +  4·t4  −  4·t5  +  4·t6  −  2·t7  +  t8

  ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)

• The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
• The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 8 minimal generators of maximal degree 8:

1. a_2_0, a nilpotent element of degree 2
2. a_3_1, a nilpotent element of degree 3
3. a_3_0, a nilpotent element of degree 3
4. c_4_0, a Duflot element of degree 4
5. a_7_3, a nilpotent element of degree 7
6. a_7_2, a nilpotent element of degree 7
7. c_8_2, a Duflot element of degree 8
8. c_8_1, a Duflot element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 4 "obvious" relations:
   a_3_0², a_3_1², a_7_2², a_7_3²

Apart from that, there are 16 minimal relations of maximal degree 16:

1. a_2_0²
2. a_2_0·a_3_0
3. a_2_0·a_3_1
4. a_3_0·a_3_1 + a_2_0·c_4_0
5. a_2_0·a_7_2
6. a_2_0·a_7_3
7. a_3_0·a_7_2 + a_2_0·c_8_2
8. a_3_0·a_7_3 + a_2_0·c_8_1
9. a_3_1·a_7_2 − a_2_0·c_8_1
10. a_3_1·a_7_3 + a_2_0·c_8_2
11. c_8_2·a_3_0 − c_8_1·a_3_1 + c_4_0·a_7_3
12. c_8_2·a_3_1 + c_8_1·a_3_0 − c_4_0·a_7_2
13. a_7_2·a_7_3 + a_2_0·c_4_0·c_8_2
14. c_8_2·a_7_2 + c_8_1·a_7_3 + c_4_0·c_8_1·a_3_0 − c_4_0²·a_7_2
15. c_8_2·a_7_3 − c_8_1·a_7_2 + c_4_0·c_8_1·a_3_1 − c_4_0²·a_7_3
16. c_8_2² + c_8_1² − c_4_0²·c_8_2

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 16 using the Hilbert-Poincaré criterion.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_0, an element of degree 4
  2. c_8_2, an element of degree 8
• The above filter regular HSOP forms a Duflot regular sequence.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(9,2); GF(3))

1. a_2_0 → a_1_0·a_1_1
2. a_3_1 → c_2_2·a_1_0 − c_2_1·a_1_1
3. a_3_0 → c_2_2·a_1_1 + c_2_1·a_1_0
4. c_4_0 → c_2_2² + c_2_1²
5. a_7_3 → c_2_1·c_2_2²·a_1_0 + c_2_1²·c_2_2·a_1_1
6. a_7_2 → c_2_1·c_2_2²·a_1_1 − c_2_1²·c_2_2·a_1_0
7. c_8_2 → c_2_1²·c_2_2²
8. c_8_1 → c_2_1·c_2_2³ − c_2_1³·c_2_2

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

1. a_2_0 → a_1_0·a_1_1, an element of degree 2
2. a_3_1 → c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
3. a_3_0 → c_2_2·a_1_1 + c_2_1·a_1_0, an element of degree 3
4. c_4_0 → c_2_2² + c_2_1², an element of degree 4
5. a_7_3 → c_2_1·c_2_2²·a_1_0 + c_2_1²·c_2_2·a_1_1, an element of degree 7
6. a_7_2 → c_2_1·c_2_2²·a_1_1 − c_2_1²·c_2_2·a_1_0, an element of degree 7
7. c_8_2 → c_2_1²·c_2_2², an element of degree 8
8. c_8_1 → c_2_1·c_2_2³ − c_2_1³·c_2_2, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps